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• Expectation of the sum of a random number of random variables If X = PN i=1 Xi, N is a random variable independent of Xi'sXi's have common mean µThen EX = ENµ • Example Suppose that the expected number of acci8 E(X Y) = E(X) E(Y) (The expectation of a sum = the sum of the expectations This rule extends as you would expect it to when there are more than 2 random variables, eg E(X Y Z) = E(X) E(Y) E(Z)) 9 If X and Y are independent, E(XY) = E(X)E(Y) (This rule extends as you would expect it to for more than 2 randomVi vil gjerne vise deg en beskrivelse her, men området du ser på lar oss ikke gjøre det

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Y=e^x graph-Layzix (@layzix) sur TikTok 4513K j'aime 458K fans Regarde la dernière vidéos de Layzix (@layzix)X and Y, ie corr(X,Y) = 1 ⇐⇒ Y = aX b for some constants a and b The correlation is 0 if X and Y are independent, but a correlation of 0 does not imply that X and Y are independent 33 Conditional Expectation and Conditional Variance Throughout this section, we will assume for simplicity that X and Y are discrete random variables

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E(X) is the expectation value of the continuous random variable X x is the value of the continuous random variable X P(x) is the probability mass function of X Properties of expectation Linearity When a is constant and X,Y are random variables E(aX) = aE(X) E(XY) = E(X) E(Y) Constant When c is constant E(c) = c ProductQed Example 2 edit If X {\displaystyle X} is a continuous random variable uniformly distributed on − 1 , 1 {\displaystyle 1,1} and Y = X 2 {\displaystyle Y=X^{2}} , then X {\displaystyle X} and Y {\displaystyle Y} are uncorrelated even though X {\displaystyle X} determines Y {\displaystyle Y} and a particular value of Y {\displaystyle Y} can be produced by only one or two((y squared ) divide by 2) plus (2y(e to the power of x)) plus (y plus e to the power of x)(dy divide by dx) equally 0 ((y to the power of two) divide by two) plus (2y(e to the power of x)) plus (y plus e to the power of x)(dy divide by dx) equally zero

 In this section we will discuss implicit differentiation Not every function can be explicitly written in terms of the independent variable, eg y = f(x) and yet we will still need to know what f'(x) is Implicit differentiation will allow us to find the derivative in these cases Knowing implicit differentiation will allow us to do one of the more important applications of derivativesAssumeX andY bivariatenormal,asfoundbyKarlPearsonaround1900 Assume E(X)=68(inches),E(Y)=69,σ X =σ Y =2,ρ=5 (Weexpectρtobepositive becauseontheaverage,thetallerthefather,thetallertheson) GivenX=80(6feet8inches),Y isnormalwithmean µ Y ρσ Y σ X (x−µ X)=695(80−68)=75Easy as pi (e) Unlock StepbyStep Natural Language Math Input

Take a trip into an upgraded, more organized inbox Sign in and start exploring all the free, organizational tools for your email Check out new themes, send GIFs, find every photo you've ever sent or received, and search your account faster than everFor any two random variables X and Y if X Y orE X E Y E X Y E X E Y The above from MATH 3603 at HKUDepartment of Computer Science and Engineering University of Nevada, Reno Reno, NV 557 Email Qipingataolcom Website wwwcseunredu/~yanq I came to the US

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Extended Keyboard Examples Upload Random Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals For mathTumblr is a place to express yourself, discover yourself, and bond over the stuff you love It's where your interests connect you with your peopleThe fact that some of the coeffi­ cients are functions of x should not slow us down Applying the quadratic formula we get y = ex ± (−ex)2 − 4 1 (−ex) 2 1 ex± √ 2 4 y = 2 Our original equation is

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Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the yaxis It looks as if you may have reached the right integral, which is \int_0^1 2\pi xe^ {x^2}\, dx You can quickly integrate by making the substitution u=x^2In addition to cookies that are strictly necessary to operate this website, we use the following types of cookies to improve your experience and our services Functional cookies to enhance your experience (eg remember settings), Performance cookies to measure the website's performance and improve your experience, Advertising/Targeting cookies, which are set by third parties withIf y = ex, then ln(y) = x or If w = ln(x), then ew = x Before we go any further, let's review some properties of this function ln(x 1x 2) = ln x 1 ln x 2 ln1 = 0 ln e = 1 These can be derived from the definition of ln x as the inverse of the function ex, the definition of e, and the rules of exponents we reviewed at the start of lecture

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Characteristic functions I Let X be a random variable I The characteristic function of X is de ned by ˚(t) = ˚ X(t) = EeitX I Recall that by de nition eit = cos(t) i sin(t) I Characteristic function ˚ X similar to moment generating function M X I ˚ XY = ˚ X˚ Y, just as M XY = M XM Y, if X and Y are independent I And ˚ aX(t) = ˚ X(at) just as M aX(t) = M X(at)Y x ce 2 x e x from y 0 c 1 3 hence c 2 and y x 2 e 2 y ( x ) = Ce 2 x e x From y (0) = C 1 = 3 Hence C = 2 and y ( x ) = 2 e 2 x e x Example 8 Given the IVP y 0 y = f ( x ) , y (0) = 2 , I = (0 , ∞ ) 8 where f ( x ) = 1 , 0 ≤ x ≤ 1 0 , x > 1 f is discontinuous at x = 1 From e R dx = e x , we get e x y 0 e x y = dY=e^ (32*x) WolframAlpha Area of a circle?

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Solution to Differential equations question Solve the following first order linear differential equationsy'y=(3x^24x3)e^x⃤ Plainmath is a free databaseModel Y provides maximum versatility—able to carry 7 passengers and their cargo Each second row seat folds flat independently, creating flexible storage for skis, furniture, luggage and more The liftgate opens to a low trunk floor that makes loading and unloading easy and quick Versatile seating and storage for cargo and passengersSearch the world's information, including webpages, images, videos and more Google has many special features to help you find exactly what you're looking for

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Y = a e^(b x) where a and b are constants The curve that we use to fit data sets is in this form so it is important to understand what happens when a and b are changed Recall that any number or variable when raised to the 0 power is 1 In this case if b or x is 0 then, e^0 = 1A Asus4 Well, let's steal a spaceship and head for the sun A Asus4 Shoot the stars with a lemonade ray gun F Make a movie and a TV show E You be Jane, I'm George Jetson Chorus A I am you and you are me A X, Y, Zee to A, B, Cee G A You, me, us, we are one A I am you and you are me A X, Y, Zee to A, B, Cee G A You, me, us, we are oneEnjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on

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Characterizations The six most common definitions of the exponential function exp(x) = e x for real x are 1 Define e x by the limit = → () 2 Define e x as the value of the infinite series = =!SOLUTION 1 Begin with x 3 y 3 = 4 Differentiate both sides of the equation, getting D ( x 3 y 3) = D ( 4 ) , D ( x 3) D ( y 3) = D ( 4 ) , (Remember to use the chain rule on D ( y 3) ) 3x 2 3y 2 y' = 0 , so that (Now solve for y' ) 3y 2 y' = 3x 2, and Click HERE to return to the list of problems SOLUTION 2 Begin with (xy) 2 = x y 1 Differentiate both sidesRead It Watch It Submit Answer 2 Points DETAILS SCALCET9 (a) Find the differential dy y = 24 x2 dy = (b) Evaluate dy for the given values of x Question (a) Find the differential dy y = e*/5 dy = x (b) Evaluate dy for the given values of x and dx x = 0, dx = 005 dy = = Enhanced Feedback Please try again

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Given the equation $$\left(e^{x} 1\right) y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} e^{y{\left(x \right)}} = 0$$ This differential equation has the formX Y Z z = Z X Y Z • xyz =1 Third component can always be computed from first two • Typically, (x,y)are specified • Letαbeanypositiveconstant Then(X,Y,Z)and(αX,αY,αZ) have the same chromaticity coordinates • Projectionproperty StraightlinesinXYZ maptostraight lines in (x,y) y The chain rule for this case is, dz dt = ∂f ∂x dx dt ∂f ∂y dy dt d z d t = ∂ f ∂ x d x d t ∂ f ∂ y d y d t So, basically what we're doing here is differentiating f f with respect to each variable in it and then multiplying each of these by the derivative of that variable with respect to t t

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The product rule for derivatives states that given a function #f(x) = g(x)h(x)#, the derivative of the function is #f'(x) = g'(x)h(x) g(x)h'(x)# The product rule is used primarily when the function for which one desires the derivative is blatantly the product of two functions, or when the function would be more easily differentiated if looked at as the product of two functions1m Followers, 284 Following, Posts See Instagram photos and videos from A͙͛H͙͛N͙͛ E͙͛L͙͛L͙͛Y͙͛ (@x_xellybabyx)Example 5 X and Y are jointly continuous with joint pdf f(x,y) = (e−(xy) if 0 ≤ x, 0 ≤ y 0, otherwise Let Z = X/Y Find the pdf of Z The first thing we do is draw a picture of the support set (which in this case is the first

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Note that if f(X;Y) is a linear combination of Xand Y, this result matches the wellknown result from mathematical statistics that E(aXbY) = aEXbEY = a x bSteps for Solving Linear Equation y = \frac { x y } { 1 x } y = y = 1 x x y y = Variable x cannot be equal to 1 since division by zero is not defined Multiply both sides of the equation by x1 Variable x cannot be equal to − 1 since division by zero is not definedY 2 − e x y − e x = 0 This is a second degree polynomial in y;

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Denotes the factorial of nOne proof that e is irrational uses a special case of this formula) 3 Define e x to be the unique number y > 0 such thatX Y E olve the problem 8) A croissant, a cup of coffee, and a fruit bowl from Kelley's Coffee Cart cost a total of $525 Kelley posts a notice announcing that, effective next week, the price of a croissant will go up 25% and the price of coffee will go up 35%Theorem 36 Let F be any partition of the set S Define a relation on S by x R y iff there is a set in F which contains both x and y Then R is an equivalence relation and the equivalence classes of R are the sets of F Pf Since F is a partition, for each x in S there is one (and only one) set of F

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